Geodesic
Commutativity of the Centralizer of a Subalgebra in a~Universal Enveloping Algebra
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 47-63.

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Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic zero, and let $\mathfrak{h}$ be an algebraic subalgebra of the tangent Lie algebra $\mathfrak{g}$ of $G$. We find all subalgebras $\mathfrak h$ that have no nontrivial characters and whose centralizers $\mathfrak{U}(\mathfrak{g})^\mathfrak{h}$ and $P(\mathfrak{g})^{\mathfrak{h}}$ in the universal enveloping algebra $\mathfrak{U}\mathfrak{g})$ and in the associated graded algebra $P(\mathfrak{g})$, respectively, are commutative. For all these subalgebras, we prove that ${\mathfrak U}\mathfrak{(g)}^{\mathfrak h}=\mathfrak{U(h)^h}\otimes\mathfrak{U(g)^g}$ and $P\mathfrak{(g)}^{\mathfrak h}=P\mathfrak{(h)^h}\otimes P\mathfrak{(g)^g}$. Furthermore, we obtain a criterion for the commutativity of $\mathfrak{U(g)^h}$ in terms of representation theory.
Keywords: universal enveloping algebra, centralizer of algebra
Mots-clés : Poisson algebra, coisotropic action. 
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A. A. Zorin. Commutativity of the Centralizer of a Subalgebra in a~Universal Enveloping Algebra. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 47-63. http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a3/

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